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The lattice shear $(a,b)\mapsto (a+mb,b)$ shifts the chain charts by $m$ and preserves $q$. On the dense torus it acts as $(t,q)\mapsto (q^{m}t,q)$. For $m\ge 3$, translates of each chart of the formal completion of a scheme along $q=0$ are disjoint. Glue $m$ representative formal charts cyclically by the adjacent-chart formulas to obtain a formal scheme quotient. The two branch opens of each chart are disjoint because $D(x_i)\cap D(y_i)=D(q)$ is empty on the formal support. The quotient fibre is an $m$-gon of projective lines, and the quotient map is locally an isomorphism on each translated chart. Invariant morphisms descend uniquely through this gluing.

// Target: ringed-space.bigb

Restricting the topological structure sheaf of a formal completion of a scheme to an open subscheme commutes with its inverse limit: both sheaves are ${\underleftarrow{\lim }}_n({\mathcal{O}}_U/(\mathcal{I}{|}_U{)}^{n+1})$ on $|U\cap Y|$. Equivalently, the identical infinitesimal neighbourhoods have identical restricted structure sheaves. On formal affine charts this uses completed localization of an adic ring.

// Target: toric-geometry.bigb

A formal scheme is a locally ringed space with a sheaf of topological rings that is locally a formal spectrum of a complete adic ring. An ordinary scheme is the special case of the discrete topology. A formal completion of a scheme retains all infinitesimal neighbourhoods of a closed subscheme, rather than just that subscheme.

// Target: ringed-space.bigb

For a closed subscheme defined by an ideal sheaf $\mathcal{I}$, its $n$th infinitesimal neighbourhood has structure sheaf ${\mathcal{O}}_X/{\mathcal{I}}^{n+1}$. All these neighbourhoods have the same underlying topological space because the radicals of $\mathcal{I}$ and its positive powers agree. Their inverse limit of structure sheaves is the formal completion of a scheme.

// Target: ringed-space.bigb

Let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be the ideal sheaf of $Y$. The formal completion of a scheme along $Y$ has underlying topological space $|Y|$ and topological structure sheafThe topology is the adic topology given by the kernels of the projections to the successive infinitesimal neighbourhoods $X_n=(|Y|,{\mathcal{O}}_X/{\mathcal{I}}^{n+1})$. In the locally Noetherian scheme setting of these examples this is a formal scheme. On a Noetherian affine scheme, with $Y=V(I)\subseteq \mathrm{Spec}A$, it is the formal spectrum $\mathrm{Spf}\hat{A}$, where $\hat{A}={\underleftarrow{\lim }}_{n}A/I^{n+1}$. Thus the completion retains all orders of infinitesimal information normal to $Y$.

Here $Z=D(t)=\mathrm{Spec}k[q,t,t^{-1}]$ is an open subscheme of $X$. The relevant affine completions arewith the $q$-adic topology. The complement of $t=0$ in the first formal scheme is its formal open $D(t)$. The sections on this open are the completed localization of an adic ring:The same calculation on every smaller principal open subset identifies the structure sheaves, not just their global sections. Equivalently, every infinitesimal neighbourhood of $Z\cap (q=0)$ is precisely the restriction to $D(t)$ of the corresponding neighbourhood in $X$. By formal completion commutes with restriction to an open subscheme,The isomorphism is canonical and respects the morphisms to $\mathrm{Spf}k[[q]]$.

It would give the wrong answer to replace the formal-open structure sheaf by the ordinary localization of a ring $k[t][[q]][t^{-1}]$. That ring has a uniform bound on the negative powers of $t$ across the coefficients of any one element, whereashas no such bound. This difference between ordinary and completed localization does not obstruct the isomorphism of formal schemes; it explains why the completed localization is essential.

Write $\hat{X}=X_{/Y}$. The formal completion of a scheme restricts on each chart toThus $\hat{X}$ is a formal scheme over $\mathrm{Spf}\mathbb{Z}[[q]]$. The ordinary torus $D(q)$ has disappeared from its underlying space: every point of $\hat{X}$ lies on $Y$. In particular, only adjacent completed charts intersect, and their two possible branch overlaps inside any one chart are disjoint.

Construct the action using the integral lattice automorphismIt carries $v_i$ to $v_{i+3}$ and ${\sigma }_i$ to ${\sigma }_{i+3}$, so induces a toric morphism $g:X\to X$ that is an automorphism. It preserves the lattice projection to the second coordinate, hence preserves $q$ and $Y$. On the dense algebraic torus it isIn the local coordinates this sends $U_i$ to $U_{i+3}$ withThese identities show regularity on every chart without requiring $q$ to be invertible there. They also make the action continuous in the $q$-adic topology, so it extends to $\hat{X}$. Its powers shift $D_i$ to $D_{i+3k}$; no nonzero power is the identity or fixes a point of the formal support. Therefore $G=⟨g⟩\cong \mathbb{Z}$ is a nontrivial infinite cyclic group acting freely.

Construct the quotient explicitly, using three representative chartswith $q=X_i{Y}_i$. Glue their branch opens cyclically, taking subscripts modulo three, byThese are genuine isomorphisms of formal schemes. For example, the completed localization of an adic ring on the first open iswhere $X_i=q/Y_i$. On the second it is $\mathbb{Z}[X_{i+1},X_{i+1}^{-1}][[q]]$, and the gluing sends $X_{i+1}$ to $Y_i^{-1}$ and $Y_{i+1}$ to $q{Y}_i$. Thus it preserves $q$ and is continuous, with a continuous inverse.

Inside ${\mathfrak{V}}_i$ we haveConsequently there are no triple-overlap conditions left to check: the two branch identifications in a chart have disjoint domains. By gluing of schemes along open subschemes in its formal version, the three charts produce a formal scheme $\mathfrak{C}$ over $\mathrm{Spf}\mathbb{Z}[[q]]$. Equivalently, reduce the same gluing modulo $q^{n+1}$ for every $n$; this constructs compatible schemes $C_n$, all on the same underlying space, with $C_n=C_{n+1}\times \mathrm{Spec}\mathbb{Z}[q]/(q^{n+1})$. Their inverse system has precisely the displayed formal charts.

For every integer $i$, map ${\hat{U}}_i$ isomorphically to ${\mathfrak{V}}_{\bar{i}}$ by identifying its coordinates with the corresponding capital-letter coordinates. The adjacent-chart formulas agree with the cyclic identifications, so these maps glue toThey are invariant under $g$, and ${\pi }^{-1}({\mathfrak{V}}_i)$ is the disjoint union of the translated opens ${\hat{U}}_{i+3k}$, each mapping isomorphically onto ${\mathfrak{V}}_i$. The group permutes these copies transitively. Thus the fibres of $\pi$ are exactly the $G$-orbits, and invariant sections on this inverse image are exactly the sections on ${\mathfrak{V}}_i$.

To verify the quotient property also for morphisms, take a $G$-invariant morphism $h:\hat{X}\to \mathfrak{T}$. Its restrictions to ${\hat{U}}_0,{\hat{U}}_1,{\hat{U}}_2$ agree on the adjacent branch opens. On the closing overlap they agree because the translated chart ${\hat{U}}_3$ has the same restriction as ${\hat{U}}_0$ under $g$. Hence they descend to a unique morphism $\mathfrak{C}\to \mathfrak{T}$. This proves that the gluing is the cyclic formal quotient of the infinite toric chain:Its fibre modulo $q$ is a triangle of three projective lines over $\mathbb{Z}$, each meeting its two neighbours at its two boundary sections. More generally, a shift by any $m\ge 3$ gives an $m$-gon. Completion is important here: translates of a formal chart are disjoint, whereas all ordinary toric charts still share the generic torus.